Fantasy Land Specification

(aka "Algebraic JavaScript Specification")

This project specifies interoperability of common algebraic structures:

• Setoid
• Ord
• Semigroupoid
• Category
• Semigroup
• Monoid
• Group
• Filterable
• Functor
• Contravariant
• Apply
• Applicative
• Alt
• Plus
• Alternative
• Foldable
• Traversable
• Chain
• ChainRec
• Monad
• Extend
• Comonad
• Bifunctor
• Profunctor

General

An algebra is a set of values, a set of operators that it is closed under and some laws it must obey.

Each Fantasy Land algebra is a separate specification. An algebra may have dependencies on other algebras which must be implemented.

Terminology

1. "value" is any JavaScript value, including any which have the structures defined below.
2. "equivalent" is an appropriate definition of equivalence for the given value. The definition should ensure that the two values can be safely swapped out in a program that respects abstractions. For example:
   • Two lists are equivalent if they are equivalent at all indices.
   • Two plain old JavaScript objects, interpreted as dictionaries, are equivalent when they are equivalent for all keys.
   • Two promises are equivalent when they yield equivalent values.
   • Two functions are equivalent if they yield equivalent outputs for equivalent inputs.

Type signature notation

The type signature notation used in this document is described below:¹

• :: "is a member of".
  • e :: t can be read as: "the expression e is a member of type t".
  • true :: Boolean - "true is a member of type Boolean".
  • 42 :: Integer, Number - "42 is a member of type Integer and Number".
• New types can be created via type constructors.
  • Type constructors can take zero or more type arguments.
  • Array is a type constructor which takes one type argument.
  • Array String is the type of all arrays of strings. Each of the following has type Array String: [], ['foo', 'bar', 'baz'].
  • Array (Array String) is the type of all arrays of arrays of strings. Each of the following has type Array (Array String): [], [ [], [] ], [ [], ['foo'], ['bar', 'baz'] ].
• Lowercase letters stand for type variables.
  • Type variables can take any type unless they have been restricted by means of type constraints (see fat arrow below).
• -> (arrow) Function type constructor.
  • -> is an infix type constructor that takes two type arguments where left argument is the input type and the right argument is the output type.
  • ->'s input type can be a grouping of types to create the type of a function which accepts zero or more arguments. The syntax is: (<input-types>) -> <output-type>, where <input-types> comprises zero or more comma–space (, )-separated type representations and parens may be omitted for unary functions.
  • String -> Array String is a type satisfied by functions which take a String and return an Array String.
  • String -> Array String -> Array String is a type satisfied by functions which take a String and return a function which takes an Array String and returns an Array String.
  • (String, Array String) -> Array String is a type satisfied by functions which take a String and an Array String as arguments and return an Array String.
  • () -> Number is a type satisfied by functions which do not take arguments and return a Number.
• ~> (squiggly arrow) Method type constructor.
  • When a function is a property of an Object, it is called a method. All methods have an implicit parameter type - the type of which they are a property.
  • a ~> a -> a is a type satisfied by methods on Objects of type a which take a type a as an argument and return a value of type a.
• => (fat arrow) Expresses constraints on type variables.
  • In a ~> a -> a (see squiggly arrow above), a can be of any type. Semigroup a => a ~> a -> a adds a constraint such that the type a must now satisfy the Semigroup typeclass. To satisfy a typeclass means to lawfully implement all functions/methods specified by that typeclass.

For example:

traverse :: Applicative f, Traversable t => t a ~> (TypeRep f, a -> f b) -> f (t b)
'------'    '--------------------------'    '-'    '-------------------'    '-----'
 '           '                               '      '                        '
 '           ' - type constraints            '      ' - argument types       ' - return type
 '                                           '
 '- method name                              ' - method target type

---

1. See the Types section in Sanctuary's docs for more info. ↩

Prefixed method names

In order for a data type to be compatible with Fantasy Land, its values must have certain properties. These properties are all prefixed by fantasy-land/. For example:

//  MyType#fantasy-land/map :: MyType a ~> (a -> b) -> MyType b
MyType.prototype['fantasy-land/map'] = ...

Further in this document unprefixed names are used just to reduce noise.

For convenience you can use fantasy-land package:

var fl = require('fantasy-land')

// ...

MyType.prototype[fl.map] = ...

// ...

var foo = bar[fl.map](x => x + 1)

Type representatives

Certain behaviours are defined from the perspective of a member of a type. Other behaviours do not require a member. Thus certain algebras require a type to provide a value-level representative (with certain properties). The Identity type, for example, could provide Id as its type representative: Id :: TypeRep Identity.

If a type provides a type representative, each member of the type must have a constructor property which is a reference to the type representative.

Algebras

Setoid

1. a.equals(a) === true (reflexivity)
2. a.equals(b) === b.equals(a) (symmetry)
3. If a.equals(b) and b.equals(c), then a.equals(c) (transitivity)

equals method

equals :: Setoid a => a ~> a -> Boolean

A value which has a Setoid must provide an equals method. The equals method takes one argument:

a.equals(b)

1. b must be a value of the same Setoid

   1. If b is not the same Setoid, behaviour of equals is unspecified (returning false is recommended).

2. equals must return a boolean (true or false).

Ord

A value that implements the Ord specification must also implement the Setoid specification.

1. a.lte(b) or b.lte(a) (totality)
2. If a.lte(b) and b.lte(a), then a.equals(b) (antisymmetry)
3. If a.lte(b) and b.lte(c), then a.lte(c) (transitivity)

lte method

lte :: Ord a => a ~> a -> Boolean

A value which has an Ord must provide a lte method. The lte method takes one argument:

 a.lte(b)

1. b must be a value of the same Ord

   1. If b is not the same Ord, behaviour of lte is unspecified (returning false is recommended).

2. lte must return a boolean (true or false).

Semigroupoid

1. a.compose(b).compose(c) === a.compose(b.compose(c)) (associativity)

compose method

compose :: Semigroupoid c => c i j ~> c j k -> c i k

A value which has a Semigroupoid must provide a compose method. The compose method takes one argument:

a.compose(b)

1. b must be a value of the same Semigroupoid

   1. If b is not the same semigroupoid, behaviour of compose is unspecified.

2. compose must return a value of the same Semigroupoid.

Category

A value that implements the Category specification must also implement the Semigroupoid specification.

1. a.compose(C.id()) is equivalent to a (right identity)
2. C.id().compose(a) is equivalent to a (left identity)

id method

id :: Category c => () -> c a a

A value which has a Category must provide an id function on its type representative:

C.id()

Given a value c, one can access its type representative via the constructor property:

c.constructor.id()

1. id must return a value of the same Category

Semigroup

1. a.concat(b).concat(c) is equivalent to a.concat(b.concat(c)) (associativity)

concat method

concat :: Semigroup a => a ~> a -> a

A value which has a Semigroup must provide a concat method. The concat method takes one argument:

s.concat(b)

1. b must be a value of the same Semigroup

   1. If b is not the same semigroup, behaviour of concat is unspecified.

2. concat must return a value of the same Semigroup.

Monoid

A value that implements the Monoid specification must also implement the Semigroup specification.

1. m.concat(M.empty()) is equivalent to m (right identity)
2. M.empty().concat(m) is equivalent to m (left identity)

empty method

empty :: Monoid m => () -> m

A value which has a Monoid must provide an empty function on its type representative:

M.empty()

Given a value m, one can access its type representative via the constructor property:

m.constructor.empty()

1. empty must return a value of the same Monoid

Group

A value that implements the Group specification must also implement the Monoid specification.

1. g.concat(g.invert()) is equivalent to g.constructor.empty() (right inverse)
2. g.invert().concat(g) is equivalent to g.constructor.empty() (left inverse)

invert method

invert :: Group g => g ~> () -> g

A value which has a Group must provide an invert method. The invert method takes no arguments:

g.invert()

1. invert must return a value of the same Group.

Filterable

1. v.filter(x => p(x) && q(x)) is equivalent to v.filter(p).filter(q) (distributivity)
2. v.filter(x => true) is equivalent to v (identity)
3. v.filter(x => false) is equivalent to w.filter(x => false) if v and w are values of the same Filterable (annihilation)

filter method

filter :: Filterable f => f a ~> (a -> Boolean) -> f a

A value which has a Filterable must provide a filter method. The filter method takes one argument:

v.filter(p)

1. p must be a function.

   1. If p is not a function, the behaviour of filter is unspecified.
   2. p must return either true or false. If it returns any other value, the behaviour of filter is unspecified.

2. filter must return a value of the same Filterable.

Functor

1. u.map(a => a) is equivalent to u (identity)
2. u.map(x => f(g(x))) is equivalent to u.map(g).map(f) (composition)

map method

map :: Functor f => f a ~> (a -> b) -> f b

A value which has a Functor must provide a map method. The map method takes one argument:

u.map(f)

1. f must be a function,

   1. If f is not a function, the behaviour of map is unspecified.
   2. f can return any value.
   3. No parts of f's return value should be checked.

2. map must return a value of the same Functor

Contravariant

1. u.contramap(a => a) is equivalent to u (identity)
2. u.contramap(x => f(g(x))) is equivalent to u.contramap(f).contramap(g) (composition)

contramap method

contramap :: Contravariant f => f a ~> (b -> a) -> f b

A value which has a Contravariant must provide a contramap method. The contramap method takes one argument:

u.contramap(f)

1. f must be a function,

   1. If f is not a function, the behaviour of contramap is unspecified.
   2. f can return any value.
   3. No parts of f's return value should be checked.

2. contramap must return a value of the same Contravariant

Apply

A value that implements the Apply specification must also implement the Functor specification.

1. v.ap(u.ap(a.map(f => g => x => f(g(x))))) is equivalent to v.ap(u).ap(a) (composition)

ap method

ap :: Apply f => f a ~> f (a -> b) -> f b

A value which has an Apply must provide an ap method. The ap method takes one argument:

a.ap(b)

1. b must be an Apply of a function

   1. If b does not represent a function, the behaviour of ap is unspecified.
   2. b must be same Apply as a.

2. a must be an Apply of any value

3. ap must apply the function in Apply b to the value in Apply a

   1. No parts of return value of that function should be checked.

4. The Apply returned by ap must be the same as a and b

Applicative

A value that implements the Applicative specification must also implement the Apply specification.

1. v.ap(A.of(x => x)) is equivalent to v (identity)
2. A.of(x).ap(A.of(f)) is equivalent to A.of(f(x)) (homomorphism)
3. A.of(y).ap(u) is equivalent to u.ap(A.of(f => f(y))) (interchange)

of method

of :: Applicative f => a -> f a

A value which has an Applicative must provide an of function on its type representative. The of function takes one argument:

F.of(a)

Given a value f, one can access its type representative via the constructor property:

f.constructor.of(a)

1. of must provide a value of the same Applicative

   1. No parts of a should be checked

Alt

A value that implements the Alt specification must also implement the Functor specification.

1. a.alt(b).alt(c) is equivalent to a.alt(b.alt(c)) (associativity)
2. a.alt(b).map(f) is equivalent to a.map(f).alt(b.map(f)) (distributivity)

alt method

alt :: Alt f => f a ~> f a -> f a

A value which has a Alt must provide a alt method. The alt method takes one argument:

a.alt(b)

1. b must be a value of the same Alt

   1. If b is not the same Alt, behaviour of alt is unspecified.
   2. a and b can contain any value of same type.
   3. No parts of a's and b's containing value should be checked.

2. alt must return a value of the same Alt.

Plus

A value that implements the Plus specification must also implement the Alt specification.

1. x.alt(A.zero()) is equivalent to x (right identity)
2. A.zero().alt(x) is equivalent to x (left identity)
3. A.zero().map(f) is equivalent to A.zero() (annihilation)

zero method

zero :: Plus f => () -> f a

A value which has a Plus must provide an zero function on its type representative:

A.zero()

Given a value x, one can access its type representative via the constructor property:

x.constructor.zero()

1. zero must return a value of the same Plus

Alternative

A value that implements the Alternative specification must also implement the Applicative and Plus specifications.

1. x.ap(f.alt(g)) is equivalent to x.ap(f).alt(x.ap(g)) (distributivity)
2. x.ap(A.zero()) is equivalent to A.zero() (annihilation)

Foldable

1. u.reduce is equivalent to u.reduce((acc, x) => acc.concat([x]), []).reduce

reduce method

reduce :: Foldable f => f a ~> ((b, a) -> b, b) -> b

A value which has a Foldable must provide a reduce method. The reduce method takes two arguments:

u.reduce(f, x)

1. f must be a binary function

   1. if f is not a function, the behaviour of reduce is unspecified.
   2. The first argument to f must be the same type as x.
   3. f must return a value of the same type as x.
   4. No parts of f's return value should be checked.

2. x is the initial accumulator value for the reduction

   1. No parts of x should be checked.

Traversable

A value that implements the Traversable specification must also implement the Functor and Foldable specifications.

1. t(u.traverse(F, x => x)) is equivalent to u.traverse(G, t) for any t such that t(a).map(f) is equivalent to t(a.map(f)) (naturality)

2. u.traverse(F, F.of) is equivalent to F.of(u) for any Applicative F (identity)

3. u.traverse(Compose, x => new Compose(x)) is equivalent to new Compose(u.traverse(F, x => x).map(x => x.traverse(G, x => x))) for Compose defined below and any Applicatives F and G (composition)

var Compose = function(c) {
  this.c = c;
};

Compose.of = function(x) {
  return new Compose(F.of(G.of(x)));
};

Compose.prototype.ap = function(f) {
  return new Compose(this.c.ap(f.c.map(u => y => y.ap(u))));
};

Compose.prototype.map = function(f) {
  return new Compose(this.c.map(y => y.map(f)));
};

traverse method

traverse :: Applicative f, Traversable t => t a ~> (TypeRep f, a -> f b) -> f (t b)

A value which has a Traversable must provide a traverse method. The traverse method takes two arguments:

u.traverse(A, f)

1. A must be the type representative of an Applicative.

2. f must be a function which returns a value

   1. If f is not a function, the behaviour of traverse is unspecified.
   2. f must return a value of the type represented by A.

3. traverse must return a value of the type represented by A.

Chain

A value that implements the Chain specification must also implement the Apply specification.

1. m.chain(f).chain(g) is equivalent to m.chain(x => f(x).chain(g)) (associativity)

chain method

chain :: Chain m => m a ~> (a -> m b) -> m b

A value which has a Chain must provide a chain method. The chain method takes one argument:

m.chain(f)

1. f must be a function which returns a value

   1. If f is not a function, the behaviour of chain is unspecified.
   2. f must return a value of the same Chain

2. chain must return a value of the same Chain

ChainRec

A value that implements the ChainRec specification must also implement the Chain specification.

1. M.chainRec((next, done, v) => p(v) ? d(v).map(done) : n(v).map(next), i) is equivalent to (function step(v) { return p(v) ? d(v) : n(v).chain(step); }(i)) (equivalence)
2. Stack usage of M.chainRec(f, i) must be at most a constant multiple of the stack usage of f itself.

chainRec method

chainRec :: ChainRec m => ((a -> c, b -> c, a) -> m c, a) -> m b

A Type which has a ChainRec must provide a chainRec function on its type representative. The chainRec function takes two arguments:

M.chainRec(f, i)

Given a value m, one can access its type representative via the constructor property:

m.constructor.chainRec(f, i)

1. f must be a function which returns a value
   1. If f is not a function, the behaviour of chainRec is unspecified.
   2. f takes three arguments next, done, value
      1. next is a function which takes one argument of same type as i and can return any value
      2. done is a function which takes one argument and returns the same type as the return value of next
      3. value is some value of the same type as i
   3. f must return a value of the same ChainRec which contains a value returned from either done or next
2. chainRec must return a value of the same ChainRec which contains a value of same type as argument of done

Monad

A value that implements the Monad specification must also implement the Applicative and Chain specifications.

1. M.of(a).chain(f) is equivalent to f(a) (left identity)
2. m.chain(M.of) is equivalent to m (right identity)

Extend

A value that implements the Extend specification must also implement the Functor specification.

1. w.extend(g).extend(f) is equivalent to w.extend(_w => f(_w.extend(g)))

extend method

extend :: Extend w => w a ~> (w a -> b) -> w b

An Extend must provide an extend method. The extend method takes one argument:

 w.extend(f)

1. f must be a function which returns a value

   1. If f is not a function, the behaviour of extend is unspecified.
   2. f must return a value of type v, for some variable v contained in w.
   3. No parts of f's return value should be checked.

2. extend must return a value of the same Extend.

Comonad

A value that implements the Comonad specification must also implement the Extend specification.

1. w.extend(_w => _w.extract()) is equivalent to w (left identity)
2. w.extend(f).extract() is equivalent to f(w) (right identity)

extract method

extract :: Comonad w => w a ~> () -> a

A value which has a Comonad must provide an extract method on itself. The extract method takes no arguments:

w.extract()

1. extract must return a value of type v, for some variable v contained in w.
   1. v must have the same type that f returns in extend.

Bifunctor

A value that implements the Bifunctor specification must also implement the Functor specification.

1. p.bimap(a => a, b => b) is equivalent to p (identity)
2. p.bimap(a => f(g(a)), b => h(i(b)) is equivalent to p.bimap(g, i).bimap(f, h) (composition)

bimap method

bimap :: Bifunctor f => f a c ~> (a -> b, c -> d) -> f b d

A value which has a Bifunctor must provide a bimap method. The bimap method takes two arguments:

c.bimap(f, g)

1. f must be a function which returns a value

   1. If f is not a function, the behaviour of bimap is unspecified.
   2. f can return any value.
   3. No parts of f's return value should be checked.

2. g must be a function which returns a value

   1. If g is not a function, the behaviour of bimap is unspecified.
   2. g can return any value.
   3. No parts of g's return value should be checked.

3. bimap must return a value of the same Bifunctor.

Profunctor

A value that implements the Profunctor specification must also implement the Functor specification.

1. p.promap(a => a, b => b) is equivalent to p (identity)
2. p.promap(a => f(g(a)), b => h(i(b))) is equivalent to p.promap(f, i).promap(g, h) (composition)

promap method

promap :: Profunctor p => p b c ~> (a -> b, c -> d) -> p a d

A value which has a Profunctor must provide a promap method.

The profunctor method takes two arguments:

c.promap(f, g)

1. f must be a function which returns a value

   1. If f is not a function, the behaviour of promap is unspecified.
   2. f can return any value.
   3. No parts of f's return value should be checked.

2. g must be a function which returns a value

   1. If g is not a function, the behaviour of promap is unspecified.
   2. g can return any value.
   3. No parts of g's return value should be checked.

3. promap must return a value of the same Profunctor

Derivations

When creating data types which satisfy multiple algebras, authors may choose to implement certain methods then derive the remaining methods. Derivations:

• equals may be derived from lte:

  function(other) { return this.lte(other) && other.lte(this); }

• map may be derived from ap and of:

  function(f) { return this.ap(this.of(f)); }

• map may be derived from chain and of:

  function(f) { return this.chain(a => this.of(f(a))); }

• map may be derived from bimap:

  function(f) { return this.bimap(a => a, f); }

• map may be derived from promap:

  function(f) { return this.promap(a => a, f); }

• ap may be derived from chain:

  function(m) { return m.chain(f => this.map(f)); }

• reduce may be derived as follows:

  function(f, acc) {
    function Const(value) {
      this.value = value;
    }
    Const.of = function(_) {
      return new Const(acc);
    };
    Const.prototype.map = function(_) {
      return this;
    };
    Const.prototype.ap = function(b) {
      return new Const(f(b.value, this.value));
    };
    return this.traverse(x => new Const(x), Const.of).value;
  }

• map may be derived as follows:

  function(f) {
    function Id(value) {
      this.value = value;
    }
    Id.of = function(x) {
      return new Id(x);
    };
    Id.prototype.map = function(f) {
      return new Id(f(this.value));
    };
    Id.prototype.ap = function(b) {
      return new Id(this.value(b.value));
    };
    return this.traverse(x => Id.of(f(x)), Id.of).value;
  }

• filter may be derived from of, chain, and zero:

  function(pred) {
    var F = this.constructor;
    return this.chain(x => pred(x) ? F.of(x) : F.zero());
  }

• filter may be derived from concat, of, zero, and reduce:

  function(pred) {
    var F = this.constructor;
    return this.reduce((f, x) => pred(x) ? f.concat(F.of(x)) : f, F.zero());
  }

If a data type provides a method which could be derived, its behaviour must be equivalent to that of the derivation (or derivations).

Notes

1. If there's more than a single way to implement the methods and laws, the implementation should choose one and provide wrappers for other uses.
2. It's discouraged to overload the specified methods. It can easily result in broken and buggy behaviour.
3. It is recommended to throw an exception on unspecified behaviour.
4. An Identity container which implements many of the methods is provided by sanctuary-identity.

Alternatives

There also exists Static Land Specification with exactly the same ideas as Fantasy Land but based on static methods instead of instance methods.